IOP PUBLISHING
NANOTECHNOLOGY
Nanotechnology 20 (2009) 035706 (6pp)
doi:10.1088/0957-4484/20/3/035706
Elastic moduli of faceted aluminum nitride
nanotubes measured by contact resonance
atomic force microscopy
G Stan1 , C V Ciobanu2 , T P Thayer2 , G T Wang3 , J R Creighton3 ,
K P Purushotham4, L A Bendersky5 and R F Cook1
1
Ceramics Division, National Institute of Standards and Technology, Gaithersburg,
MD 20899, USA
2
Division of Engineering, Colorado School of Mines, Golden, CO 80401, USA
3
Advanced Materials Sciences Department, Sandia National Laboratories, Albuquerque,
NM 87185, USA
4
Precision Engineering Division, National Institute of Standards and Technology,
Gaithersburg, MD 20899, USA
5
Metallurgy Division, National Institute of Standards and Technology, Gaithersburg,
MD 20899, USA
E-mail: gheorghe.stan@nist.gov and cciobanu@mines.edu
Received 25 August 2008, in final form 24 October 2008
Published 17 December 2008
Online at stacks.iop.org/Nano/20/035706
Abstract
A new methodology for determining the radial elastic modulus of a one-dimensional
nanostructure laid on a substrate has been developed. The methodology consists of the
combination of contact resonance atomic force microscopy (AFM) with finite element analysis,
and we illustrate it for the case of faceted AlN nanotubes with triangular cross-sections. By
making precision measurements of the resonance frequencies of the AFM cantilever-probe first
in air and then in contact with the AlN nanotubes, we determine the contact stiffness at different
locations on the nanotubes, i.e. on edges, inner surfaces, and outer facets. From the contact
stiffness we have extracted the indentation modulus and found that this modulus depends
strongly on the apex angle of the nanotube, varying from 250 to 400 GPa for indentation on the
edges of the nanotubes investigated.
(Some figures in this article are in colour only in the electronic version)
actuators, and nano-electromechanical systems (NEMS) [7, 8].
Given the diversity in morphology, cross-section, and size
resulting from current synthesis methods [1–6, 9, 10], the
properties of one-dimensional (1D) AlN nanostructures that
are important for the fabrication and performance of NEMS,
in particular mechanical properties, can be rather difficult
to quantify. This difficulty is due to size-dependent effects
at the nanoscale, large surface-to-volume ratio, and the
possible presence of defects acquired during the nanostructure
growth process. Furthermore, in the current indentationbased methods used to measure the elastic properties of
AlN bulk single crystals (e.g., [11]) the sensing indentation
depths are on the order of hundreds nanometers. This means
that such methods cannot be used to measure the elastic
1. Introduction
Aluminum nitride has an intriguing combination of physical
properties, such as enhanced field emission, large optical
band gap, high thermal conductivity, large electrical resistivity,
as well as a piezoelectric coefficient comparable to that of
quartz. These properties make AlN nanostructures suitable
for advanced nanoscale electronic and optoelectronic device
applications, and have motivated sustained efforts to synthesize
AlN nanostructures in various morphologies: wires [1, 2],
nanoparticles [3], nanotubes [4], needles [5], and platelets [6].
While many applications for AlN nanostructures target their
use as field emitters in flat panel displays, their superior
piezoelectric properties and integration compatibility with
silicon substrates make them excellent candidates for sensors,
0957-4484/09/035706+06$30.00
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© 2009 IOP Publishing Ltd Printed in the UK
Nanotechnology 20 (2009) 035706
G Stan et al
properties of AlN nanostructures, critical to NEMS design
and performance, that have diameters comparable to the
typical indentation depths. It is therefore necessary to design
procedures to accurately determine the elastic properties of
1D AlN structures, particularly methods that can locally probe
the variation of elastic moduli along an axis and in various
locations on a cross-section.
Here, we report experimental measurements of local
elastic properties of faceted AlN nanotubes (AlN NTs) with
triangular cross-section by contact resonance atomic force
microscopy (CR-AFM) [12, 13]. The AlN NTs were obtained
from core–shell GaN–AlN nanowires after thermally removing
the GaN cores, which is a novel procedure for synthesizing
NTs with controllable wall thickness [14]. We characterized
the AlN NTs using a suite of microscopy techniques to
determine their crystallography and geometric dimensions.
Next, we performed CR-AFM measurements on AlN NTs
to determine the stiffness of the contact formed between
the atomic force microscope (AFM) tip and the NTs. As
the contact stiffness depends on both the elastic properties
of the materials and the contact geometry, it cannot be
readily mapped onto an indentation modulus value of an AlN
NT. We therefore performed finite element analysis (FEA)
calculations in order to account for the actual experimental
contact geometry and to convert the contact stiffness into a
unique value for the indentation modulus of the NT. Due to
the combination with FEA simulations for realistic geometries,
our procedure to determine the elastic properties breaks away
from the usual analysis based on Hertzian contact models
and is applicable in general for 1D nanostructures of any
cross-sectional shape, potentially yielding the radial elastic
modulus at any location along a substrate-supported nanowire
or nanotube. Such local probing gives accurate values of the
elastic moduli of 1D nanostructures, and can also reveal the
presence of defects via variations of these moduli.
Figure 1. (a)–(c) SEM images of faceted AlN nanotubes, in lateral
(a) and axial views ((b), (c)). The images clearly show that the
nanotube cross-sections are triangular, with the side of the triangles
of the order of 200 nm. (d) TEM image of an AlN NT grown along
the [1̄21̄0] direction. The thickness of the nanotube walls was
estimated at 19 ± 1 nm. From the corresponding SAED pattern
(shown in inset), distances between crystalline planes were
determined to be 0.25 nm and 0.15 nm for the (0002) and the (1̄21̄0)
families of planes, respectively. These distances are consistent with
the AlN lattice constants c = 0.498 and a = 0.311 nm [17].
patterns, which show that the NTs were predominately single
crystals of wurtzite structure with the 1̄21̄0 growth direction
(figure 1(d)). The measured lattice parameters fit well the
wurtzite structure of AlN [17]. Observed intensity streaks in
the [0001] direction of [011̄0] SAED pattern in the inset of
figure 1(d) suggest that one of the tube facets is a (0001) plane.
Consequently, the facets forming the triangular tubular prisms
are the planes (0001), (101̄1), and (1̄011). This configuration
is similar to that of GaN nanowires [15, 18] grown under
similar conditions to the removed GaN cores in our work.
The wall thickness of the NTs was measured as 19 ± 1 nm
from the thickness fringes that shadow the tube walls in TEM
images. A wall thickness of 20 ± 1 nm was measured on tube
walls exposed parallel to the electron beam in SEM. The SEM
images were taken with a FEI Helios dual-beam microscope6
at an accelerating voltage of 15 kV and a beam current of
86 pA. The TEM images were acquired on a JEOL JEM-3010
transmission electron microscope at 300 kV.
2. AlN nanotubes: synthesis and morphology
The AlN NTs in our study were synthesized through an
‘epitaxial casting’ process similar to that reported previously
for the synthesis of GaN nanotubes [14]. Using triangularfaceted GaN nanowires synthesized by Ni-catalyzed metal–
organic chemical vapor deposition (MOCVD) [15] at 800 ◦ C,
AlN shells were grown around the GaN nanowires by
MOCVD at 1000 ◦ C [16]. The GaN cores were subsequently
removed by annealing in hydrogen atmosphere at 1120 ◦ C
for approximately 3 h, leaving behind the empty AlN shells.
The AlN NTs so obtained were harvested and dispersed onto
Au films, Cu grids, and Si substrates for various microscopy
measurements. The structure and morphology of AlN NTs
were investigated by means of scanning electron microscopy
(SEM), transmission electron microscopy (TEM), and AFM.
Each technique indicated that the AlN NTs were hollow prisms
with rounded edges that preserved the triangular cross-section
of the GaN core templates. Such triangular cross-sections are
shown in SEM images in figures 1(a)–(c). The crystallographic
structure of the NTs was established from TEM images and
their corresponding selected-area electron diffraction (SAED)
3. CR-AFM measurements on AlN nanotubes
To quantify the elastic properties of the AlN NTs, we have
employed the CR-AFM technique [12, 13] adapted for 1D
nanostructures axially supported on surfaces [19, 20]. In this
6
Certain commercial equipment, instruments, or materials are identified
in this document. Such identification does not imply recommendation or
endorsement by the National Institute of Standards and Technology, nor does
it imply that the products identified are necessary the best available for the
purpose.
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Nanotechnology 20 (2009) 035706
G Stan et al
Figure 2. (a) AFM image of a long AlN NT. (b) Detailed AFM profile of the 500 nm nanotube segment highlighted in (a). (c) Averaged AFM
trace (red curve) of the profile shown in (b), which was used to extract the shape of the AlN NT via an erosional fit (green curve). The
resulting nanotube profile (black curve) is characterized by 60◦ basal angles and a rounded apex with a radius of 20 nm.
technique, the stiffness k of the contact formed between an
AFM tip and a nanotube is determined from the shift in the
cantilever’s resonance frequencies measured when the tip is
brought from air into contact with the AlN NT. The contact
stiffness is then used to calculate the indentation modulus
of the NT sample, MNT , by means of the Hertz contact
formula [21]:
MNT Mt
k = 2a
,
(1)
MNT + Mt
where a is the contact radius, and MNT and Mt are the
indentation moduli of the NT sample and the tip, respectively.
The indentation moduli for isotropic materials have simple
expressions in terms of Young’s modulus E and Poisson’s
2
ratio ν , e.g., MNT = E NT /(1 − νNT
); anisotropy corrections
to the indentation modulus may also be evaluated [23, 22].
Equation (1) implies that in order to calculate the indentation
modulus MNT from the contact stiffness k it is necessary to
have knowledge of the contact geometry, i.e. of the contact
radius a .
CR-AFM measurements were performed on a commercial
AFM (Veeco MultiMode III, Santa Barbara CA) with
additional instrumentation (LabView, National Instruments,
Austin, TX) (see footnote 6). A lock-in amplifier with internal
signal generator (7280 Signal Recovery AMETEK, Oak Ridge,
TN) was used to induce vibrations in the AFM cantilever
and to collect the high-frequency signal from the photodiode
detector [35]. The AFM probes (PPP-SEIH NanoSensors,
Neuchatel, Switzerland) used in this work were single-crystal
Si cantilevers with [100]-oriented tips. The cantilever stiffness
kc was determined to be 11.5 ± 0.5 N m−1 by the thermalnoise method [36]. A constant applied load F of about 200 nN
was applied in each measurement. The applied load (kept
constant throughout the measurements) was much greater that
the adhesion force between the tip and surface while small
enough that the contact deformations are in the linear elastic
regime.
Taking advantage of the spatial resolution allowed by
CR-AFM, we have made elastic modulus measurements at
Figure 3. Thickness of AlN NT walls determined from AFM
measurements of a nanotube sectioned along its axis. (a) AFM image
of a micrometer-long nanotube. (b) Detailed profile of the end
portion of the tube (dashed-line contour in (a)). (c) Line profile
obtained with the tip moved in the axial plane highlighted in (b);
the wall thickness is found to be 20 ± 1 nm.
different locations on AlN NTs: on top edges of a few tubes
laid down on the substrate (one of these measured tubes is
shown in figure 2), on an outer nanotube facet positioned
parallel to the substrate (not shown here), and on the inside
facet of a broken tube (shown in figure 3). The parameters
necessary to resolve the geometry of the contact between
the tip and the round top edge of the prismatic NTs were
determined from topographical AFM scans of the AlN NTs.
Figures 2(a) and (b) show detailed 3D AFM topographies
of a sample prismatic AlN nanotube. The average crosssection of the AFM trace shown in figure 2(b) was used to
construct the outer profile of the nanotube in figure 2(c) by
subtracting the contribution of the tip from the AFM trace.
The angles formed by the side facets of the AlN NT with the
substrate are not affected by the tip’s contribution to the AFM
image (figure 2(c)). For the case shown in figure 2(c) these
basal angles were found to be 60◦ , indicating a cross-section
similar to that formed by the crystallographic planes (0001),
(101̄1), and (1̄011) of the wurtzite structure. Deviations less
3
Nanotechnology 20 (2009) 035706
G Stan et al
Figure 5. Geometric configuration of the contact between
(a) tip-on-flat Si(100) surface and (b) tip-on-prismatic edge of AlN
nanotube. Under the same applied force, the contact area is circular
in (a) and elliptical in (b). (c) Schematic geometry used in FEA
calculations. The points on the x –z support plane of the triangular
nanotube have z -component of the displacements set to zero, and a
restraint is placed on the hemispherical tip to prevent it from slipping
along the nanotube surface. To speed up calculations, we exploit the
symmetry of the system about the y –z plane by setting to zero the
x -displacements of the points on this plane.
geometry and to extract the elastic modulus MNT from the
contact stiffness kNT
is detailed below.
Figure 4. The shifts of (a) first and (b) second resonance frequencies
measured when the AFM tip is brought from air into contact with a
Si(100) reference surface (black curves), with the inner surface of a
sectioned tube (green curves), and with an outer facet of AlN NTs
(orange curves).
4. Elastic modulus of faceted AlN nanotubes
Using the geometry of the nanotube and the radius of the AFM
tip derived from the AFM trace (refer to figure 2), FEA was
performed to determine the load, F , versus displacement, δ ,
curve for radial indentation of the AlN NTs. We have used
the FEA package Nastran/Patran (MSC Software Corp.) and
solved the equations of elasticity with an adaptive mesh to
improve convergence. We have started the FEA calculations
with uniform meshes that have 3000–5000 tetragonal elements
and up to 10 000 nodes. During FEA calculations, the adaptive
mesh becomes finer in the high-stress contact region and
coarser far away form it, as shown in figure 5(c). For clarity
of the visualization, in figure 5(c) we show an intermediate
stage of the calculation where it is apparent that the mesh
has become denser in the contact region. We have replaced
the substrate that supports the nanowire with rigid boundary
conditions (refer to figure 5(c)), which is justified because,
for the loads considered here, stresses decay rapidly with the
distance from the contact point and become negligible in the
vicinity of the supporting substrate. The tip was constrained to
move only vertically, so as not to slip on the nanotube during
the FEA calculation. Lastly, in order to speed up the solution
we have taken advantage of the symmetry across the y –z plane
(figure 5(c)) by setting the lateral displacements (i.e. along the
x direction) to zero on that plane.
The displacement of the tip in the FEA calculations was
recorded for each value of the applied load. For loads smaller
than 500 nN and indentation moduli of the AlN NT assumed
to be in the range 200 GPa < MNT < 500 GPa, FEA
calculation showed that the displacement of the AFM tip in
contact with the AlN NT is proportional to F 2/3 ; apart from
multiplicative factors that depend on the nanotube geometry
and elastic modulus, this is the same dependence as that for
Hertzian contact [21]. Consequently, the contact stiffness
(defined as kNT
= ∂ F/∂δ ) is proportional to F 1/3 . As the
than 10◦ were observed for the facet angles for all AlN NTs
investigated. The round top edge observed in AFM scans (see
figure 2), had a curvature radius determined to be 20 nm.
For each tested AlN NT, the contact stiffness kNT
was
calculated (in terms of the cantilever stiffness kc ) from the first
two contact resonance frequencies, f 1,NT and f 2,NT , measured
in five CR-AFM frequency sweeps. Figure 4 shows the
shifts in CR-AFM frequencies when the cantilever was moved
from air into contact with the AlN NTs flat facets, as well
as onto a Si(100) reference surface. By tuning the applied
load, we have ensured an increased sensitivity of the contact
resonance frequencies to variations in the elastic properties of
the materials tested. For the resonances shown in figure 4, the
system behaved as a clamped-spring coupled beam rather than
a clamped–clamped beam, i.e., the ratio of the first contact
resonance frequency to the first free resonance frequency of
the cantilever was smaller than 4 (this ratio is about 5 in the
case of clamped–clamped cantilever beam) [12]. On a given
NT sample, the average values of the two contact resonance
frequencies were used in a clamped-spring coupled beam
model [12] of the cantilever to uniquely determine the contact
stiffness kNT
. The geometry of the contact plays a significant
role in extracting the elastic modulus of the sample tested from
the contact stiffness. In particular, a substantial change in the
contact area occurs when CR-AFM measurements on the top
edge of prismatic AlN nanotubes have to be calibrated to those
made on a flat Si(100) surface. The contact geometries for the
tip in contact with the reference Si(100) surface and the tip
on the nanotube edge are shown schematically in figures 5(a)
and (b), respectively. The procedure to account for the contact
4
Nanotechnology 20 (2009) 035706
G Stan et al
Table 1. Indentation modulus MNT for selected AlN NTs, as
determined by CR-AFM measurements with the AFM tip in contact
with the nanotube apex. The NTs have isosceles triangle
cross-sections with rounded apices, and are characterized by the
angle α of a lateral facet with the substrate and the height h . The last
two rows of the table show the indentation modulus measured on an
inner and outer flat surfaces of the AlN nanotubes.
Angle α (deg)
Height h (nm)
MNT (GPa)
60.0
60.5
62.5
64.5
66.5
69.0
Inner NT facet
Flat NT facet
115
88
104
198
181
127
274.0 ± 29.5
279.5 ± 18.1
306.2 ± 23.7
253.3 ± 21.7
378.5 ± 36.8
372.1 ± 38.9
325.2 ± 15.5
305.0 ± 12.7
and table 1) had the greatest indentation modulus, indicative
of a size-dependent effect in which this modulus increases
with decreasing apex angle at the rounded edge of the AlN
NT. However, variations in the cross-sectional dimensions
(sides, angles) of the NTs prevent us from making a more
definite connection between MNT and the apex angle; the wall
thickness and the apex radius would necessarily enter into any
size-dependence of MNT . Another factor that could contribute
to variations of the measured MNT is the roughness on the core
surfaces, which has been observed on the GaN nanowires [16].
Since the core has rough surfaces, it is expected that the
AlN shell has rough or defected surfaces too. To release the
mismatch strain between AlN and GaN, threading dislocations
may also form at the core–shell interface during the growth
of the shell as observed in thin-film systems [24]. While
some of these dislocations may be annealed away during the
core removal, it is unlikely for the remaining AlN shell to be
dislocation-free.
Interestingly, despite the presence of possible defects, the
range of measured MNT values includes the theoretical values
for the indentation modulus of bulk single-crystal AlN of
300 GPa (indentation perpendicular to c-axis) and 335 GPa
(indentation parallel to c-axis), which have been calculated
using the elastic constants for AlN given in [25]. We also
determined the stiffness ratio associated with the flat surfaces
of the NTs, which can expose an inner side (figure 3) or
an outer facet parallel to the substrate (not shown). For the
indentation of flat facets, the FEA curve can be replaced by
the exact analytic formula from the Hertz model (dashed-line
curve in figure 6). The indentation moduli of AlN NT facets
were found to be 305 GPa (outer facets) and 325 GPa (inner
side), in good agreement with the theoretical values listed
above. The indentation modulus values for flat facets are also
included in table 1. In contrast to the case of nanowires, where
the elastic modulus depends on their diameter [19, 20, 26, 27],
no significant deviation from bulk value was observed here for
the out-of-plane modulus of AlN facets as thin as 20 nm.
Figure 6. Determination of the indentation modulus MNT of AlN
nanotubes from CR-AFM measurements. The shifts in resonance
frequencies set the ratio kNT
/kSi
(100) (e.g., red solid dot on the vertical
axis), which used in conjunction with the FEA-calculated
dependence (solid curve) gives the indentation modulus of that
nanotube (i.e. the MNT coordinate of the open circles). A similar
procedure has been applied for the contact between the AFM tip and
flat surfaces of the AlN NTs, for which we have used the Hertz
contact model formula [21] (dashed curve) instead of the
FEA-calculated dependence.
stiffness kSi
(100) of the contact between the AFM tip and the flat
Si(100) surface (taken as Hertzian) is also proportional to F 1/3 ,
it follows that the ratio kNT
/kSi
(100) is independent of load and
can thus be plotted as a function of the indentation modulus of
the NT. This FEA dependence is shown in figure 6 for a typical
set of geometric parameters, i.e. tip radius of 15 nm, basal
angle of the nanotube facet of 60◦ , nanotube wall thickness
of 20 nm, and radius of curvature at the top of the AlN NT
of 20 nm. These values are representative for our experiment
and allow us to draw conclusions about the elastic properties
of AlN NTs. FEA calculations can yield extensive families
of curves for different tip radii, NT basal angles and wall
thicknesses, and such families of curves can be used to refine
the elastic moduli results; this refinement, however, is only
necessary for nanotubes with much smaller radial dimensions
than those synthesized in our experiments. For a given elastic
modulus, the stiffness of the tip–tube contact always lies below
of the stiffness of the tip–flat contact (figure 6) because of the
increased structural compliance of the tube compared to that
of a flat slab of the same size. The FEA stiffness ratio curve
is the key to estimating the elastic modulus of the AlN NT,
because a given stiffness ratio kNT
/kSi
(100) as determined from
resonance frequency measurements corresponds, via the FEA
curve, to a unique value for the indentation modulus of the AlN
NT (figure 6).
Using the FEA dependence, we have found that the
indentation modulus MNT for the rounded edges of the AlN
NTs falls in the range 250–400 GPa, depending on the
geometrical parameters of the tubes investigated. We note
that the NTs with the greatest basal angle (refer to figure 6
5. Conclusions
We have synthesized AlN NTs by an epitaxial casting
process in which AlN was deposited on GaN triangular cores.
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Nanotechnology 20 (2009) 035706
G Stan et al
The cores were subsequently removed by annealing in a
hydrogen atmosphere, and the resulting AlN NTs have been
characterized by SEM, AFM, and TEM. We have investigated
the radial elastic moduli of the AlN NTs laid down on a
substrate using a combination of CR-AFM measurements
and FEA calculations.
Our approach complements a
class of techniques for probing elastic properties of 1D
nanostructures, which include in situ electromechanical
resonance [28], bending tests [29–31], static and dynamic
nanoindentation [32, 33], and tensile stress measurements [34].
The CR-AFM technique is based on measuring the shifts
in resonant frequencies when the AFM tip is brought from
air in contact with the sample, and from these frequency
shifts we have calculated the stiffness associated with the
contact between the AFM tip and nanotube tested. The FEA
calculations enable us to use the contact stiffness to extract
the indentation modulus of the AlN NTs for different contact
geometries. The values that we determined for indentation
modulus depend on the nanotube dimensions, but remain
close to those of bulk AlN (300–335 GPa) [25] for most
NTs investigated. For AlNTs with 20 nm thick walls, the
indentation moduli corresponding to the inner surfaces and
outer facets are determined to be 325 GPa and 305 GPa,
respectively. Therefore, we have found that the reduction in
size (a requirement for NEMS applications) does not imply a
significant distortion of the mechanical properties of AlN NTs
compared to those of bulk AlN. It is conceivable, however,
that a moderate variation in the mechanical properties of AlN
NTs synthesized through the ‘epitaxial casting’ process may
be achieved by selecting GaN NWs templates with different
cross-sections and different effective diameters.
We note that the approach presented here is general, and
can be used for nanowires or nanotubes made of any material
to determine the radial indentation modulus at any location on
their outer surface. The only prerequisite for such general use
is the ability to correctly characterize the shape and dimensions
of the cross-section, which can be achieved straightforwardly
by AFM for nanostructures as thin as 10 nm in diameter.
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Acknowledgments
G T Wang acknowledges support from the DOE Office of
Basic Energy Sciences and Sandia’s Laboratory Directed
Research and Development program. Sandia is a multiprogram
laboratory operated by Sandia Corporation, a Lockheed Martin
Company, for the United States Department of Energy’s
National Nuclear Security Administration under contract
No. DE-AC04-94Al85000.
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